Where Is My Material Textures Influence Panel?
Shortly after I posted this, I narrowed down the answer!

After tracking down the differences between each set up one by one I realized I had forgotten one alteration: PC A's version of Blender had the Luxrender addon. After a very brief search I discovered this Luxrender bug.

Deactivating Luxrender and reopening Blender did the trick. The influence panel is back!

Thanks to anybody who came by. Maybe this will help someone else.